reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem Th15:
    for p be Polynomial of n,R holds Zero_{p} = Zero_(p)
    proof
      let p be Polynomial of n,R;
      set S = {p};
      thus for o holds o in Zero_S implies o in Zero_(p)
      proof
        let o;
        assume o in Zero_S; then
        o in {x where x is Function of n,R : for f be Polynomial of n,R
        st f in S holds eval(f,x) = 0.R} by Def6; then
        consider x be Function of n,R such that
A3:     x = o &
        for f be Polynomial of n,R st f in S holds eval(f,x) = 0.R;
        assume not o in Zero_(p); then
A5:     eval(p,x) <> 0.R by A3;
        p in S by TARSKI:def 1;
        hence contradiction by A5,A3;
      end;
      thus for o holds o in Zero_(p) implies o in Zero_S
      proof
        let o;
        assume
        o in Zero_(p); then
        consider x1 be Function of n,R such that
A6:     x1 = o & eval(p,x1) = 0.R;
        for f be Polynomial of n,R st f in S holds eval(f,x1) = 0.R
        by A6,TARSKI:def 1; then
        x1 in {x where x is Function of n,R : for f be Polynomial of n,R
        st f in S holds eval(f,x) = 0.R};
        hence thesis by A6,Def6;
      end;
    end;
